User:Oh Isaac/Advanced Mechanics of Materials and Applied Elasticity

Equation of Advanced Mechanics of Materials and Applied Elasticity
REDIRECT User:Oh_Isaac

three-dimensional state of stress

 * $$[\tau_{i,j}]=

\begin{bmatrix} \sigma_x & \tau_{x,y} & \tau_{x,z} \\ \tau_{y,x} & \sigma_y & \tau_{y,z} \\ \tau_{z,x} & \tau_{z,y} & \sigma_z \end{bmatrix} $$

prismatic bars of linearly elastic material
\tau=\frac{T\rho}{J} $$ \sigma_x=-\frac{My}{I} $$ \tau=\frac{VQ}{Ib}$$
 * axial loading $$\sigma_x=\frac{P}{A}$$
 * torsion $$
 * bending $$
 * shear $$
 * $$T$$ torque.
 * $$V$$ vertical shear force from bending force.
 * $$I$$ moment of inertia about neutral axis(N.A.).
 * $$J$$ polar moment of inertia of circular cross section.
 * $$\rho$$ distance from the center of torque to the point.
 * $$Q$$ first moment about N.A. of the area beyond the point at which \tau_{x,y} is calculated.

thin-walled pressure vessels

 * cylinder
 * circular $$

\sigma_\theta=\frac{pr}{t}. $$
 * axial $$\sigma_a=\frac{pr}{2t}.

$$ \sigma=\frac{pr}{2t}. $$
 * sphere $$
 * $$\sigma_\theta$$ tangential stress in cylinder wall.
 * $$\sigma_a$$ axial stress in cylinder wall.
 * $$\sigma$$ membrane stress in sphere wall.
 * $$p$$ internal pressure.
 * $$t$$ wall thickness.
 * $$r$$ mean radius.

axially loaded members
$$\sigma_{x'}=\sigma_x\cos ^2\theta.$$

$$\tau_{x'y'}=-\sigma_x\sin \theta\cos \theta$$

$$\sigma_{max}=\sigma_x$$

$$\tau_{max}=\pm0.5\sigma_x$$
 * $$\theta_{\max{\sigma}}=0^\circ,180^\circ. $$


 * $$\rho_{\max{\sigma}}=45^\circ,135^\circ. $$

differential equations of equilibrium
$$\frac{\partial \tau_{i,j}}{\partial x_j}+F_i=0, i,j=x,y,z.$$

plane-stress transformation
(2-dimensional stress, neglect the stress in the z coordinate.)

$$\sigma_{x'}=\tfrac{1}{2}(\sigma_x+\sigma_y)+\tfrac{1}{2}(\sigma_x-\sigma_y)\cos 2\theta+\tau_{xy}\sin 2\theta$$

$$\tau_{x'y'}=-\tfrac{1}{2}(\sigma_x-\sigma_y)\sin 2\theta+\tau_{xy}\cos 2\theta$$

$$\sigma_{y'}=\tfrac{1}{2}(\sigma_x+\sigma_y)-\tfrac{1}{2}(\sigma_x-\sigma_y)\cos 2\theta-\tau_{xy}\sin 2\theta$$

Stress tensor

$$\sigma_{x'}+\sigma_{y'}=\sigma_x+\sigma_y=$$constant.

$$\theta_{\min}=31.7^\circ+90^\circ(+180^\circ).$$

$$\theta_{\max}=31.7^\circ(+180^\circ).$$

$$\tau_{\min}=31.7^\circ(+90^\circ).$$

$$\tau_{\max}=31.7^\circ+45^\circ(+90^\circ).$$

principal stresses in plane
$$\sigma_{\max,\min}=\sigma_{1,2}=\frac{\sigma_x+\sigma_y}{2}\pm\sqrt{(\frac{\sigma_x-\sigma_y}{2})^2+\tau^2_{xy}}$$

$$\tau_{\max}=\pm\tfrac{1}{2}(\sigma_1-\sigma_2)$$

$$\tau'=\tau_{ave}=\tfrac{1}{2}(\sigma_1-\sigma_2)$$

principal stress in three dimensions
Define principal stress $$\sigma_1>\sigma_2>\sigma_3.$$

$$ \begin{vmatrix} \sigma_x-\sigma_p & \tau_{xy} & \tau_{xz} \\ \tau_{xy} & \sigma_y-\sigma_p & \tau_{xz} \\ \tau_{xz} & \tau_{xy} & \sigma_z-\sigma_p \end{vmatrix}=0 $$


 * So, get$$\sigma_p\to\sigma_1,\sigma_2,\sigma_3.$$,

and then,

$$ \begin{vmatrix} \sigma_x-\sigma_1 & \tau_{xy} & \tau_{xz} \\ \tau_{xy} & \sigma_y-\sigma_1 & \tau_{xz} \\ \tau_{xz} & \tau_{xy} & \sigma_z-\sigma_1 \end{vmatrix} \begin{pmatrix} l_1 \\ m_1\\ n_1 \end{pmatrix}=0 $$

Get $$l_1,m_1,n_1$$, continue to get

$$ A=\begin{vmatrix} l_1 & m_1 & n_1 \\ l_2 & m_2 & n_2 \\ l_3 & m_3 & n_3 \end{vmatrix} = \begin{vmatrix} \cos (x',x)& cos (x',y)& cos (x',z)\\ \cos (y',x)& cos (y',y)& cos (y',z)\\ \cos (z',x)& cos (x',y)& cos (x',z) \end{vmatrix}=0 $$

Or use $$\sigma'=A\sigma A^T$$

Calculate the main stress

$$\sigma^3_p-I_1\sigma^2_p+I_2\sigma_p-I_3=0$$

in which

$$ \begin{align} I_1 &=\sigma_x+\sigma_y+\sigma_z\\ I_2 &=\begin{vmatrix} \sigma_{1,1}&\sigma_{1,2}\\ \sigma_{2,1}&\sigma_{2,2} \end{vmatrix}+ \begin{vmatrix} \sigma_{2,2}&\sigma_{2,3}\\ \sigma_{3,2}&\sigma_{3,3} \end{vmatrix}+ \begin{vmatrix} \sigma_{1,1}&\sigma_{1,3}\\ \sigma_{3,1}&\sigma_{3,3} \end{vmatrix}=\sigma_x\sigma_y+\sigma_y\sigma_z+\sigma_x\sigma_z-\tau_{xy}^2-\tau_{yz}^2-\tau_{xz}^2\\ I_3 &=\begin{vmatrix} \sigma_x&\tau_{xy}&\tau_{xz}\\ \tau_{xy}&\sigma_y&\tau_{yz}\\ \tau_{xz}&\tau_{yz}&\sigma_z \end{vmatrix} \end{align} $$

normal and shear stresses on an oblique plane
$$\sigma_v=\sqrt{\sigma^2_{(v)1}+\sigma^2_{(v)2}+\sigma^2_{(v)3}}$$ in which
 * $$\begin{align}

\sigma_{(v)1}&=\sigma_xl+\tau_{xy}m+\tau_{xz}n\\ \sigma_{(v)2}&=\tau_{xy}l+\sigma_ym+\tau_{yz}n\\ \sigma_{(v)3}&=\tau_{xz}l+\tau_{yz}m+\sigma_zn \end{align}$$

$$\sigma_n=\sigma_xl^2+\sigma_ym^2+\sigma_zn^2+2\tau_{xy}lm+2\tau_{yz}mn+2\tau_{x}ln$$

$$\tau=\sqrt{\sigma_v^2-\sigma_n^2}$$

octahedral stresses

 * $$l=m=n=\frac{1}{\sqrt{3}}.$$
 * $$\tau_{oct}=\tfrac{1}{3}[(\sigma_1-\sigma_2)^2+(\sigma_2-\sigma_3)^2+(\sigma_3-\sigma_1)^2]^\tfrac{1}{2}.$$
 * $$\sigma_{oct}=\tfrac{1}{3}(\sigma_1+\sigma_2+\sigma_3).

$$